Exercise 6
Dose linearity and dose proportionality
Objectives of the exercise
- To learn what dose linearity is vs. dose proportionality
- To document dose proportionality using ANOVA
- To test dose linearity/proportionality by linear regression
- To test and estimate the degree of dose proportionality using a power model and a bioequivalence approach
Overview
In drug development, it is essential to determine whether the pharmacokinetic parameters of a new drug candidate, for a given dose range, are linear or nonlinear.
While a complete PK profile for all doses is impossible to establish, prediction of drug exposure in a certain dose range can easily be made if the compound possesses the property of dose proportionality (DP). This property is also called dose independent and is related to linear PK.
Linearity of drug disposition implies that all PK variables describing the drug disposition are actually parameters (constant whatever the dose).
For a linear pharmacokinetic system, the measures of exposure, such as maximal blood concentration (Cmax) or area under the curve from 0 to infinity (AUC), are proportional to the dose.
This can be expressed mathematically as:
Eq.1
where is a proportionality constant greater than zero. If the dependent variable is AUC then is the clearance of the drug that should be a constant over a range of doses.
If linear pharmacokinetics does not hold, then nonlinear pharmacokinetics is occurring, which means that measures of exposure increase in a disproportionate manner with increasing dose.
Assessment of dose proportionality (DP)
There are many ways to assess for DP (see Gough et al., 1995, PDF file available).
The three approaches are:
- Analysis of variance (ANOVA) of the PK response, normalized (divided) by dose
- Linear regression (simple linear model or model with a quadratic component)
- Power model
Linear regression
The classical approach to test DP is first to fit the PK dependent variable (AUC, Cmax) to a quadratic polynomial of the form:
Eq.2
Where the hypothesis is whether beta2 and alpha are equal or not to zero.
Dose non-proportionality is indicated if either parameter is significantly different from zero.
If only beta2 is not significantly different from 0, the simple linear regression is accepted.
Eq.3
Using equation 3, alpha is tested for zero equality. If alpha equals zero, then Eq. 1 holds and dose proportionality is declared. If alpha does not equal zero, then dose linearity (which is distinct from dose proportionality) is declared.
The main drawback of this regression approach is the lack of a measure that can quantify DP. In addition, when the quadratic term is significant or when the intercept is significant but close to zero, we are unable to estimate the magnitude of departure from DP. This point is addressed with the power model.
Analysis of variance
The second most popular approach to analyse DP is to use ANOVA.
Before analysis, correction of the dependent variable (AUC, Cmax) is made by dividing the response by the dose.
If the response is proportional to the dose, such an adjustment will make the corrected response constant, and the model is:
Eq. 4
where j is the index for dose level, µ denotes the grand mean and alpha denotes the treatment effect.
If the null hypothesis (alpha1=alpha2=alpha3….=0), is not rejected, then DP is claimed.
This model ignores the order of the doses and it is difficult to predict the response for a dose not actually tested.
Power model
An empirical relationship between AUC and dose (or C) is the following power model:
Eq.5
In this model, the exponent (beta) is a measure of the DP.
Taking the LN-transformation leads to a linear equation and the usual linear regression can then applied to this situation:
Eq.6
where beta, the slope, measures the proportionality between Dose and Y.
If beta=0, it implies that the response is independent from the dose and when beta=1, DP can be declared.
A test for dose proportionality is then to test whether beta=1. The advantage of this method is that the usual assumptions regarding homoscedasticity often apply and alternative variance models are unnecessary.
Smith et al. (2000) (Smith BP, Vandenhende FR, DeSante KA, Farid NA, Welch PA, Callaghan JT, Forgue ST. 2000. Confidence interval criteria for assessment of dose proportionality. Pharm Res 17:1278–1283 –document available in your folders) argued that it is insufficient to simply test for dose proportionality using regression methods because an imprecise study may lead to large confidence intervals around the model parameters that indicate DP, but are in fact meaningless. If Y(h) and Y(l) denote the value of the dependent variable, like Cmax, at the highest (h) and lowest (l) dose tested, respectively, and the drug is Dose Proportional then:
Eq.7
where Ratio is a constant called the maximal dose ratio. Dose proportionality is declared if the ratio of the geometric means Y(h)/Y(l) is actually equal to Ratio.
It is desirable to estimate the degree of DP. Smith et al. suggested using a bioequivalence-type approach, consisting in declaring DP only if the appropriate confidence interval for ratio (r) is entirely contained within some user-defined equivalence region {alpha (the nominal statistical risk ), Theta1 (lower bound), Theta2 (upper bound)} that is based on the drug’s safety, efficacy, or registration considerations. No scientific guidelines exist for the choice of the risk alpha, Theta1 and Theta2. Although Smith et al. suggested using values given in bioequivalence guidelines of 0.10 for , 0.80, and 1.25 for Theta1 and Theta2, respectively.
The a priori acceptable confidence interval (CI) for the SLOPE (see Smith et al for explanation) is given by the following relationship:
Here 0.8 and 1.25 are the critical a priori values suggested by regulatory authorities for any bioequivalence problem after a data log transformation.
Hence, if the (1 -alpha)% confidence interval for Slope is entirely contained within the a priori equivalence region, then dose proportionality is declared. If not, then dose non-proportionality is declared.
A working example
As an example, Table 1 presents the results obtained in mice for Bisphenol A (BPA), an endocrine disruptor.
Different doses of BPA ranging from 2.0 to 100000 µg/kg were administered by the oral route. The PK response was the plasma concentration 24h post administration and the question was to know whether or not these critical plasma concentrations were proportional to the administered dose thus concentrations were scaled by the administered dose.
Table 1: Raw data to assess dose proportionality
mouse / Concentrationng/mL / Tested dose
µg/kg / Scaled Plasma concentration (ng/mL/kg)
1 / 0.01828 / 2.3 / 7.949642355
2 / 0.00206 / 2.3 / 0.894894074
3 / 0.00477 / 2.3 / 2.075089526
4 / 0.00270 / 2.3 / 1.173539331
5 / 0.00431 / 2.3 / 1.875846129
6 / 0.00434 / 2.3 / 1.886850057
7 / 0.00396 / 2.3 / 1.721555755
8 / 0.01810 / 20.1 / 0.900697416
9 / 0.02886 / 20.1 / 1.435795812
10 / 0.04308 / 20.1 / 2.143486639
11 / 0.03399 / 20.1 / 1.690972793
12 / 0.50791 / 396.9 / 1.279704684
13 / 0.91223 / 396.9 / 2.298393056
14 / 0.42422 / 396.9 / 1.06883641
15 / 0.22045 / 396.9 / 0.555440069
16 / 0.33228 / 396.9 / 0.837196731
17 / 1.06393 / 396.9 / 2.680587657
18 / 251.76572 / 98447 / 2.557373248
19 / 67.23774 / 98447 / 0.682984124
20 / 167.39083 / 98447 / 1.700314164
21 / 157.72017 / 98447 / 1.602082054
22 / 195.69239 / 98447 / 1.987794344
Table 2 gives the arithmetic and geometric means of the dose-scaled concentrations
Dose and Conc scaled by dose2 / 20 / 400 / 100000
7.94964235 / 0.90069742 / 1.27970 / 2.55737325
0.89489407 / 1.43579581 / 2.29839 / 0.68298412
2.07508953 / 2.14348664 / 1.06884 / 1.70031416
1.17353933 / 1.69097279 / 0.55544 / 1.60208205
1.87584613 / 0.83720 / 1.98779434
1.88685006 / 2.68059
1.72155575
arithmetic mean / 2.511059604 / 1.54273817 / 1.45336 / 1.70610959
geometric mean / 1.94568608 / 1.47140508 / 1.2556 / 1.56732125
Inspection of Table 2 suggests that the mean plasma BPA concentrations, scaled by the administered dose, are rather similar across tested doses.
This is confirmed by the one way ANOVA (table3)
Table 3: ANOVA by Excel for testing dose proportionality.
Analysis of variance for dose proportionality
Analysis of variance for dose proportionalitygroups / sample number / Sum / mean / Variance
Column 1 / 7 / 17.57741723 / 2.511059604 / 5.930942299
Column 2 / 4 / 6.170952661 / 1.542738165 / 0.268841741
Column 3 / 6 / 8.720158607 / 1.453359768 / 0.716820105
Column 4 / 5 / 8.530547934 / 1.706109587 / 0.465409288
ANOVA
Source of variation / Sum of suqres / df / Mean square / F / Probability / Critical F
Entre Groupes / 4.480035277 / 3 / 1.493345092 / 0.642484468 / 0.597604381 / 3.159907598
A l'intérieur des groupes / 41.8379167 / 18 / 2.324328705
Total / 46.31795197 / 21
The P value was =0.597 indicating that the null hypothesis (dose proportionality) cannot be rejected for P<0.05.
Then the same ANOVA was performed with log-transformed data giving the same conclusion. So, a first conclusion is: “in the present experiment, when considering BPA plasma concentrations 24h post BPA dosing, there was no evidence against the null hypothesis of BPA dose proportionality for BPA doses ranging from 2.3 and 98447 µg/kg”. But be aware, this conclusion is not equivalent to a conclusion saying explicitly that “the present experiment provided evidence of dose BPA proportionality” because the fact to be unable to reject the null hypothesis is not proof of BPA dose proportionality.
Linear regression analysis
For this part of the exercise, the actual tested doses (2.3, 20.1, 396.9 and 98447 µg/kg) were replaced by the nominal doses (2.0, 20; 400; and 100000 µg/kg). Table 2 gives the WNL data sheet.
- Unweighted vs. weighted simple linear regression
All regressions were preformed using WNL. Goodness of fit was determined by visual inspection of the fittings and of the residuals' scatter plot.
The simple unweighted linear model was unacceptable due to heteroscedasticity (Figure 1). This was expected due to the very large range of tested doses. Since the variance of Y increases with increasing dose, a variance model other than constant variance needs to be used.
Figure 1: X vs. Observed Y and Predicted Y for the simple weighted linear plot;
Figure 2: X vs. Weighted (W=1) Residual Y for a simple linear model; inspection of Figure 2 indicates heteroscedasticity (i.e.an increase of data spreading for the highest dose) and results were not considered
The simple weighted (w=1/Y²) linear model including all data was also unacceptable due to a very severe misfit as shown in Figure 3.
Figure 3: X vs. Observed Y and Predicted Y for the simple weighted linear with a weighting factor W=1/Y2
The weighted polynomial regression gave also a misfit (Figure 4)
Figure 4: X vs. weighted (w=1/Y²) residual for Y for a simple linear model; inspection of Figure 4 indicates no heteroscedasticity
Thus, the simple weighted linear model including data from 2.3 to 100000µg/kg was considered because it gave an acceptable fitting (Figure 3) and an appropriate residual scatter plot (Figure 4); from this weighted linear regression approach, it can be concluded to the BPA disposition linearity for a range of BPA doses between 2 and 100 000µg/kg because the intercept is not significantly different from 0 (see Table 4).
Table 4: Data analyzed by a simple weighted (1/Y2) linear model with data corresponding to doses ranging from 2.3 to 100000µg/kg. The CI of intercept (INT) includes 0 and the intercept (alpha in Equation 1) can be considered as not different from 0. Thus BPA dose proportionality can be accepted.
However, this regression is not totally satisfactory as the different tested doses were not evenly spread and we are facing the situation where there are only 2 dose levels (low level from 2 to 400 µg/kg) and high level (100000 µg/kg). This shortcoming of the usual regression analysis will be overcome by the power model analysis.
Power model (or a linear log-log model)
Data used for the log-log linear model (LN transformation)mouse / concentration / dose( nominal) / log dose nominal / log conc
1 / 0.01828 / 2.000 / 0.693147181 / -4.001719215
2 / 0.00206 / 2.000 / 0.693147181 / -6.185896077
3 / 0.00477 / 2.000 / 0.693147181 / -5.344841858
4 / 0.00270 / 2.000 / 0.693147181 / -5.914821905
5 / 0.00431 / 2.000 / 0.693147181 / -5.44578633
6 / 0.00434 / 2.000 / 0.693147181 / -5.439937354
7 / 0.00396 / 2.000 / 0.693147181 / -5.531617765
8 / 0.01810 / 20.000 / 2.995732274 / -4.011621373
9 / 0.02886 / 20.000 / 2.995732274 / -3.545316195
10 / 0.04308 / 20.000 / 2.995732274 / -3.14460169
11 / 0.03399 / 20.000 / 2.995732274 / -3.381731483
12 / 0.50791 / 400.000 / 5.991464547 / -0.677441583
13 / 0.91223 / 400.000 / 5.991464547 / -0.091860712
14 / 0.42422 / 400.000 / 5.991464547 / -0.85750033
15 / 0.22045 / 400.000 / 5.991464547 / -1.512065482
16 / 0.33228 / 400.000 / 5.991464547 / -1.101767112
17 / 1.06393 / 400.000 / 5.991464547 / 0.061965126
18 / 251.76572 / 100000.000 / 11.51292546 / 5.528498989
19 / 67.23774 / 100000.000 / 11.51292546 / 4.208234668
20 / 167.39083 / 100000.000 / 11.51292546 / 5.120331369
21 / 157.72017 / 100000.000 / 11.51292546 / 5.060822399
22 / 195.69239 / 100000.000 / 11.51292546 / 5.276543987
Data for doses ranging from 2 to 100 000 µg/kg were analysed after a log-log transformation. The plot of the fitting is given by Figure 5 and the plot of residuals is given in Figure 6. Inspection of Figure 5 indicates a good fitting and inspection of Figure 6 indicates that the log-log transformation stabilized variance (i.e. the homoscedasticity assumption is fulfilled).
Thus the results of the regression were considered and calculated parameters are given in Table 5.
The intercept is EXP(-6.254)=0.00192; R² was of 0.99
Figure 5: Observed Y and Predicted Y for the power (linear log-log ) model with data corresponding to doses ranging from 2 to 100 000µg/kg (log-log scale); visual inspection of Figure 5 gives apparent good fitting.
Figure 6: X vs. weighted (w=1) residual Y for a log-log linear power model with data corresponding to doses ranging from 2 to 100 000µg/kg; inspection of Figure 6 indicates appropriate scatter of residuals (no bias, homoscedasticity)
Table 6: Data analyzed by a linear log-log model with doses ranging from 2 to 100000µg/kg.
Final parameters
The univariate CI for the SLOPE (0.9026-1.030) as computed by WinNonLin is a 95% CI computed with the critical ‘t’ value for 20 degree of freedom i.e. t=2.086.
To compute a 90% CI i.e. (1-2*alpha) 100%, the critical “t” for 20 ddl is 1.725 and the shortest 90% CI of the SLOPE is 0.9137-1.019; this is the classical shortest interval computed for a bioequivalence problem.
The a priori acceptable CI for the SLOPE (see Smith et al for explanation) is given by the following relationship:
Here 0.8 and 1.25 are the critical values suggested by regulatory authorities for any bioequivalence problem after a data log transformation.
The BPA dose ratio tested (higher vs. lower tested dose) was 100000/2=50000.
Thus, the a priori confidence interval for this BPA dose ratio was 0.9794-1.0206it can be concluded that both the 95 and the 90% CI for the SLOPE were not totally included in this a priori regulatory CI and then the BPA dose proportionality cannot be accepted (proved) for this range of BPA doses. As explained by Smith et al “as the dose ratio increases, the critical region for the SLOPE narrows. It is intuitive that the criterion for proportionality should be more stringent for a large dose range than that for a narrow range”.
Data for doses ranging from 2 to 400µg/kg were also analyzed after a log-log transformation. The plot of the fitting is given by Figure 7 and the plot of residuals is given in Figure 8. Inspection of Figure 7 indicates a good fitting and inspection of Figure 8 indicates that the log-log transformation stabilized variance (i.e. the homoscedasticity assumption is fulfilled).
Figure 7: Observed Y and Predicted Y for the power (linear log-log ) model with data corresponding to doses ranging from 2 to 400µg/kg (log-log scale); visual inspection of Figure 7 indicates apparent good fitting
Figure 8: X vs. Weighted (w=1) Residual Y for a log-log linear power model with data corresponding to doses ranging from 2 to 400 µg/kg; inspection of Figure 8 indicates appropriate scatter of residuals (no bias, homoscedasticity)
Thus, the results of the regression were considered and the calculated parameters are given in Table 7. Document 9 gives all the results and figures corresponding to this analysis.
Table 7:Data analyzed by a linear log-log model with doses ranging from 2 to 400µg/kg
The univariate CI for the SLOPE (0.7593-1.0212) as computed by WinNonLin is a 95% CI computed with the critical ‘t’ value for 14 ddl i.e. t=2.145.
To compute a 90% CI i.e. (1-2*alpha)100%, the critical “t” for 14 ddl is 1.761 and the shortest 90% CI of the SLOPE is 0.782-0.9984; this is the classical shortest interval computed for a bioequivalence problem.
The a priori confidence interval for this BPA dose ratio of 200 (i.e. 400/2) was 0.9628-1.0372; it can be concluded that both the 95 and the 90% CI for the SLOPE were not totally included in this a priori regulatory CI and then the BPA dose proportionality cannot be accepted for this range of BPA doses.
Conclusions
- For the entire range of tested BPA doses (2 to 100000 µg/kg) and taking into account the corresponding a priori confidence interval for the tested BPA dose ratio of 0.9794-1.0206, it can be concluded that both the 95 and the 90% CI for the SLOPE were not totally included in this a priori regulatory CI and then the BPA dose proportionality cannot be claimed (proved) for this range of BPA doses.
- From a weighted linear regression approach, the BPA linear disposition for a range of BPA doses between 2 and 100000 µg/kg can be concluded, but the range of doses is very large and the tested doses were not evenly spread, making the conclusion debatable, at best.
- When considering BPA plasma concentrations 24h post BPA dosing, an ANOVA taking into account dose-normalized concentrations provided no evidence against the null hypothesis of BPA dose proportionality for BPA doses ranging from 2.3 and 100000 µg/kg
Attached documents:
-Excel file giving the raw data and ANOVA
-Gough et al., Assessment of dose proportionality: report from the statisticians in the pharmaceutical industry. Pharmacokinetic UK Joint Working Party. Drug information Journal. 1995, 29: 1039-1040.
-Smith et al. Confidence interval criteria for assessment of dose proportionality. Pharmaceutical Research. 2000, 17: 1278-1283
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